6*x^2-8*x-3=0

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Solution for 6*x^2-8*x-3=0 equation:


Simplifying
6x2 + -8x + -3 = 0

Reorder the terms:
-3 + -8x + 6x2 = 0

Solving
-3 + -8x + 6x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-0.5 + -1.333333333x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.333333333x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.333333333x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.333333333x + x2 = 0 + 0.5
-1.333333333x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.333333333x + x2 = 0.5

The x term is -1.333333333x.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333x + 0.4444444442 + x2 = 0.5 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333x + x2 = 0.5 + 0.4444444442

Combine like terms: 0.5 + 0.4444444442 = 0.9444444442
0.4444444442 + -1.333333333x + x2 = 0.9444444442

Factor a perfect square on the left side:
(x + -0.6666666665)(x + -0.6666666665) = 0.9444444442

Calculate the square root of the right side: 0.971825316

Break this problem into two subproblems by setting 
(x + -0.6666666665) equal to 0.971825316 and -0.971825316.

Subproblem 1

x + -0.6666666665 = 0.971825316 Simplifying x + -0.6666666665 = 0.971825316 Reorder the terms: -0.6666666665 + x = 0.971825316 Solving -0.6666666665 + x = 0.971825316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = 0.971825316 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = 0.971825316 + 0.6666666665 x = 0.971825316 + 0.6666666665 Combine like terms: 0.971825316 + 0.6666666665 = 1.6384919825 x = 1.6384919825 Simplifying x = 1.6384919825

Subproblem 2

x + -0.6666666665 = -0.971825316 Simplifying x + -0.6666666665 = -0.971825316 Reorder the terms: -0.6666666665 + x = -0.971825316 Solving -0.6666666665 + x = -0.971825316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = -0.971825316 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = -0.971825316 + 0.6666666665 x = -0.971825316 + 0.6666666665 Combine like terms: -0.971825316 + 0.6666666665 = -0.3051586495 x = -0.3051586495 Simplifying x = -0.3051586495

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.6384919825, -0.3051586495}

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